Question

The value of the expression $12\left(\frac{2}{{\log }_4{\left(2000\right)}^6}+\frac{3}{{\log }_5{\left(2000\right)}^6}\right)$ is

Answer 1

$12(\frac{2}{{\log }_4{\left(2000\right)}^6}+\frac{3}{{\log }_5{\left(2000\right)}^6})=2(2{\log }_{2000}4+3{\log }_{2000}5)$ $=2{\log }_{2000}{2}^4{5}^3=2{\log }_{2000}2000=2$
$[\because \log a^m=m\log a,{\log }_{a}b=\frac{1}{{\log }_{b}a},\log (ab)=\log a+\log b,{\log }_{a}a=1]$